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190                       ANALYTICAL MECHANICS     .
sum of the increments in the kinetic energies of the individual particles.    Therefore
=2 (
When the particles form a continuous system, we can replace the particles by elements of mass and the summation sign by the integration sign. Thus
X
rs*m r2dw-~|a>02/   r2dm Jo
= ijV-i/a*2,                          (VI)
where co0 and o> are the initial and the final values of the angular velocity of the body. Therefore in this case also work done equals the increase in the kinetic energy.
PROBLEMS.
1.   The flywheel of a metal punch is 4 feet in outside diameter and weighs 500 pounds.   What must be its initial velocity in order that the punch may exert a force of 25 tons, through a distance of 1 inch, without reducing the speed of the flywheel by more than 25 per cent?   Neglect the effect of the shaft, and consider the flywheel to be a disk.
2.  The power of a 15-ton car was shut off and the brakes were put on at a time when the car was making 50 miles an hour.   On each of the 8 wheels a normal brake-shoe force of 5000 pounds was applied.   Find the distance covered by the car before coming to rest.   The diameters of the wheels are 30 inches, the tracks are horizontal, and the coefficient of friction for the contact between the shoes and the wheels is 0.2.
3.  A 100-ton locomotive making a mile a minute is to be stopped within 500 yards.   What brake-shoe force must be applied?   The diameters of the wheels are 6 feet.   The coefficient of friction is 0.3.
4.  Find the amount of heat which would be generated if the rotation of the earth about its axis were stopped.   The mean density of the earth
 5.5 -, the radius = 4000 miles; 1 calorie = 4.2 (10)7 ergs. cm*